Non linear filter with group delay at pre-response frequency for high res radio

ABSTRACT

Methods and devices are described for reducing the audible effect of pre-responses in an audio signal. The pre-responses are effectively delayed by employing a digital non-minimum-phase filter, which includes a zero lying outside the unit circle in its z-transform response. This zero is not paired with another zero at a reciprocal position inside the unit circle, as this would linearise the phase modification. The filtering can introduce a greater group delay at the pre-response frequency than at a low frequency, such as 500 Hz or even 0 Hz. The technique can be used to reduce pre-responses in an existing audio signal and also to pre-empt pre-responses that would be introduced to the audio signal by subsequent processing.

FIELD OF THE INVENTION

The invention relates to the processing of audio signals prior tocommercial distribution for improved sound quality as heard by theconsumer, and particularly to reducing the audible effect ofpre-responses.

BACKGROUND TO THE INVENTION

Until approximately 1995, the 44.1 kHz sampling rate of the Compact Disc(CD) was regarded by most people as entirely adequate. Since 1995, the‘hi-res’ movement has adopted sampling frequencies of 96 kHz, 192 kHz orhigher, potentially allowing audio bandwidths of 40 kHz, 80 kHz or more.It has always been something of a puzzle as to why there should be anyaudible advantage in the bandwidth extension, since the CD's samplingrate of 44.1 kHz allows near-perfect reproduction of audio frequenciesup to 20 kHz, the generally-accepted upper frequency limit of humanhearing.

Superior time-resolution has been advanced as a possible explanation ofthe apparent paradox, and a recent paper by J. R. Stuart and P. G.Craven “A Hierarchical Approach to Archiving and Distribution” presentedat the Audio Engineering Society Convention, Los Angeles, 11 Oct. 2014[AES preprint no. 9178], explains this concept and cites severalNeuroscience references that support this view.

According to this view, the impulse response of a recording andreproduction chain should be as compact in time as possible. Experienceindicates that audible pre-responses are particularly undesirable andthe above-cited reference presents an argument as to why this might bethe case.

The many existing recordings stored at 44.1 kHz have generally eitherbeen made using an oversampling analogue-to-digital converter providinga 44.1 kHz output, or they have been explicitly downsampled from arecording made at a higher sampling rate. Filtering is required in bothcases and until recently it was generally considered better to uselinear phase filtering. Unfortunately, linear phase filtering alwaysintroduces pre-responses.

In the case of recordings made at sample rates such as 88.2 kHz orhigher, the pre-response can be reduced by “Apodising” as described inCraven, P. G., “Antialias Filters and System Transient Response at HighSample Rates” J. Audio Eng. Soc. Volume 52 Issue 3 pp. 216-242; March2004.

Typically an 88.2 kHz sampled system will have an antialias filter thatcuts steeply at 40 kHz or some slightly higher frequency. The solutionproposed in the paper is to ‘apodise’, that is to filter more gentlystarting at 20 kHz or a slightly higher and tapering down to zero byabout 40 kHz. The sharp band-edge above 40 kHz is thereby renderedinnocuous, since the apodising filter has removed the signal energy atfrequencies that would provoke ringing or pre-responses. There remainssome pre- and/or post-response from the apodising filter itself, butthis can be much shorter in time since its transition band, from 20 kHzto 40 kHz, is much wider.

The situation is much less favourable for 44.1 kHz recordings. For theserecordings it has generally been considered ideal to use a downsamplingor antialias filter with a response flat to 20 kHz and then cuttingsharply to be essentially zero by the Nyquist frequency of 22.05 kHz. Itis thus not possible for an apodising filter to taper the responsegently to zero by the frequency of the sharp-cut filter unless theapodising filter starts to taper at a lower frequency such as 15 kHz,which is not generally considered acceptable. Sometimes it is possibleto improve the sound by a filter that begins to roll off at 20 kHz butin general there is a danger that an apodiser constrained thus willsimply replace one band-edge by another nearly as sharp and at aslightly lower frequency.

What is needed therefore is an improved or alternative technique tominimise the undesirable audible effects of pre-responses, especiallyfor signals that have been stored or will be transmitted at a relativelylow sampling rate such as 44.1 kHz.

SUMMARY OF THE INVENTION

The inventors have realised that the audibility of a pre-response can bereduced not by directly attempting to reduce the amplitude of thepre-response but rather by using a non-minimum-phase zero to introducegroup delay at frequencies where the pre-response has most energy.

Thus, according to a first aspect of the present invention, there isprovided a method of reducing the audible effect of a pre-responsehaving energy at a pre-response frequency, the method comprisingintroducing group delay at the pre-response frequency by filtering adigital audio signal using a digital non-minimum-phase filter having az-transform response that includes a zero lying outside the unit circle.

Such a zero can used to create a greater group delay at the pre-responsefrequency than at low frequencies generally, including frequencies at ornear 0 Hz. The zero should not be paired with another zero at areciprocal position inside the unit circle, as happens in conventionallinear phase filtering, as such pairing would linearise the phasemodification provided by the zero and render it ineffective as a meansof providing extra delay in the vicinity of the pre-response frequency.

A zero outside the unit circle introduces a ‘maximum phase’ element intothe filter's transfer function, the resulting group delay therebydelaying the pre-response so that its time advance to the main peak ofthe impulse response is reduced and the pre-response is thereby lessaudible. With several such zeroes acting co-operatively, thetime-advance may be reduced to zero or may be made negative; thus thepre-response may be transformed into a post-response, which is much lessaudible.

Signals retrieved from an archive may contain a pre-response already, inwhich case the invention will delay the existing pre-response.Alternatively, or in addition, the invention may be used pre-emptivelyto delay signal frequency components that could provoke the generationof a pre-response in a subsequent processing operation. In that casefiltering according to the invention will pre-emptively delay the signalfrequency components that would provoke the pre-response, delaying alsothe pre-response relative to lower-frequency components of the signal.The two situations are mathematically identical since linear filteringis a commutative operation.

Typically, pre-responses are caused by filtering operations performed inconnection with a change of sample rate, it being usual to apply asteep-cut filter at a frequency just below a ‘reference’ Nyquistfrequency corresponding to a ‘reference’ sample rate, being the lower ofthe sample rates involved. A pre-response thereby generated can beexpected to have energy that lies predominantly within 20% of the saidreference Nyquist frequency.

The method may be performed using a filter having many z-plane zeroesbut the inventors have found that often a significant audible advantagemay be obtained using a filter having as few as three zeroes outside theunit circle, each having the group delay properties referred to above.Specifically, if ‘z’ represents a time advance of one sample at asampling frequency equal to twice the reference Nyquist frequency, it ispreferred that the filter comprise at least three z-plane zeroes havingreciprocals whose real parts are each more negative than −0.5.

In some embodiments, the method of the invention will be applied to asignal that has been downsampled from a higher frequency. In that casean appropriate reference sample rate is normally the sampling frequencyof the digital audio signal.

Sometimes it is convenient to apply the method to a signal that hasalready been upsampled by a factor of two, or alternatively to a signalthat will subsequently be downsampled by a factor of two. In that case,the reference sample rate is normally one half of the sampling frequencyof the digital audio signal.

Increasingly, content is mastered for delivery at a ‘2×’ sampling ratesuch as 96 kHz but often such content is mixed from heterogeneoussources, some of which have been recorded or processed at a ‘1×’reference sample rate such as 44.1 kHz or 48 kHz. These components ofthe audio mix may contain pre-responses with energy at or just below acorresponding reference Nyquist frequency of 22.05 kHz or 24 kHz. A 96kHz sampled signal could therefore have such pre-responses along withfurther pre-responses having energy just below a signal Nyquistfrequency of 48 kHz. In such cases it may be advantageous so treat bothgroups of pre-responses according to the invention using further zeroesappropriately positioned outside the unit circle. Of course, if the ‘1×’reference sample rate is not clearly distinguished from the signalsample rate then this double processing is less relevant and it seemswise to concentrate on situations in which the ‘1×’ pre-responses have afrequency not exceeding 60% of the signal Nyquist frequency.

A z-plane zero close to a Nyquist frequency will create an amplituderesponse that is severely depressed in the vicinity of the Nyquistfrequency. The amplitude response may be flattened completely byincorporating also a pole at a reciprocal position in the z-plane, thezero and the pole in combination forming an all-pass factor in thefilter's transfer function.

Alternatively, the amplitude response may be flattened at lowerfrequencies by adding poles having frequencies slightly lower than thezeroes, the poles being configured to provide an amplitude response flatwithin a tolerance such as 1 dB over a frequency range important to theear, such as 0 to 16 kHz.

The delay produced by the filtering method of the first aspect can becharacterised by comparison with a ‘reference delay’ which could be thedelay at a lower comparison frequency such as 500 Hz or 0 Hz oralternatively it could be the delay time to the largest peak in thefilter's impulse response. Normally the delay at the pre-responsefrequency will exceed the reference delay by a finite margin, forexample by ten cycles at the pre-response frequency. For a pre-responsenear 20 kHz this would be a margin of 0.5 ms.

According to a second aspect of the present invention there is provideda mastering processor adapted to receive a first digital audio signaland to furnish a second digital audio signal for distribution, whereinthe mastering processor is configured to perform the method of the firstaspect of the invention to reduce the audible effect of a pre-responsein a signal rendered from the second signal for auditioning by alistener.

Thus, the method of the first aspect is performed by a masteringprocessor that receives audio ‘tracks’ from an archive and adjusts themprior to commercial release. Often, tracks within the archive will havepre-responses, which the method delays in order to reduce their audibleeffect. The mastering processor may also pre-emptively delaypre-responses produced by upsampling or downsampling in listeners'equipment.

According to a third aspect of the present invention there is provided aconsumer equipment having an input adapted to receive a digital audiosignal, the consumer equipment configured to process the receiveddigital audio signal according to the method of the first aspect of theinvention.

In this way, equipment designed for home listening may perform themethod of the first aspect to improve the sound quality from existingCDs and other sources that have not been mastered according to theinvention. The equipment may also perform the method in order toprecondition a digital audio signal prior to a digital-to-analogueconversion that may generate pre-responses.

It is noted that the invention may be embodied in hardware, such ascustom logic built into an ADC or DAC, or in software, or in acombination of both.

According to a fourth aspect of the present invention there is provideda recorded medium conveying a digital audio signal processed by themethod of the first aspect. Such a recording will have minimal inherentpre-responses and/or will delay the generation of pre-responses thatwould otherwise be audible on reproduction.

According to a fifth aspect of the present invention there is provided acomputer program product comprising instructions that when executed by asignal processor causes said signal processor to perform the method ofthe first aspect.

Such a program product may implement a digital signal processor (DSP)that performs the mastering behaviour of the invention. Alternatively,the program product may implement an upgrade to an existing DSP, whichallows it to perform the mastering behaviour of the invention. A similarupgrade may be provided to the processing capability of end-userconsumer equipment. Indeed, the invention may be implemented in software“apps” for mobile phones and the like, or in upgrades therefor. Theability to “retrofit” such an upgrade to existing equipment in orderthat it can implement the invention is a particularly advantageousfeature.

As will be appreciated by those skilled in the art, the presentinvention provides methods and devices for reducing the audible effectsof pre-responses in an audio signal, and which can do so in the contextof reducing the audible effects of existing pre-responses in an audiosignal and/or by taking pre-emptive action in anticipation ofpre-responses that would be introduced by subsequent processing. Thepre-responses are effectively delayed by employing a digitalnon-minimum-phase filter, which includes a zero lying outside the unitcircle in its z-transform response. Further variations andembellishments will become apparent to the skilled person in light ofthis disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

Examples of the present invention will be described in detail withreference to the accompanying drawings, in which:

FIG. 1 illustrates schematically a complete recording and reproductionchain;

FIG. 2 shows (solid line) the transition band of the filter used byAdobe ‘Audition 1.5’ when downsampling from 88.2 kHz to 44.1 kHz, and(dashed line) the transition band of an Arcam FMJ DV139 player whenplaying a CD;

FIG. 3A shows (upper trace) an impulse response of the Adobedownsampling filter of FIG. 2, the time-axis referring to sample periodsat an 88.2 kHz sampling rate, and (lower trace) an impulse response ofthe Arcam reconstruction filter of FIG. 2;

FIG. 3B shows more detail of the Adobe pre-responses of FIG. 3A;

FIG. 3C shows a spectrum of part of the Adobe pre-response of FIG. 3A

FIG. 4A shows the poles and zeroes of a 3rd order IIR low-pass filter;

FIG. 4B shows the frequency response of the filter of FIG. 4A;

FIG. 4C shows (upper trace) Arcam impulse response, and (lower trace)Arcam impulse response with pre-processing by the 3rd order IIR low-passfilter of FIG. 4A, both plotted with a 5× expanded vertical scale;

FIG. 5A shows the poles and zeroes of a maximum-phase 3rd order IIRlow-pass filter having the same frequency response as FIG. 4B;

FIG. 5B shows Arcam impulse responses with pre-processing (upper trace)by the 3rd order IIR low-pass filter of FIG. 4A, and (lower trace) bythe maximum-phase filter of FIG. 5A, both plotted with a 10× expandedvertical scale;

FIG. 6A shows the poles and zeroes of an all-pass filter having the samezeroes as the filter of FIG. 5A;

FIG. 6B shows a comparison between (upper trace) the ‘Arcam’ impulseresponse shown also in FIG. 3A; and (lower trace) the Arcam impulseresponse pre-processed by the all-pass filter of FIG. 6A, both plottedwith a 5× expanded vertical scale;

FIG. 7A shows the poles and most of the zeroes of a 12th-order all-passfilter, two of the zeroes being outside the range of the plot;

FIG. 7B shows (top trace) impulse response of Adobe ‘Audition 1.5’downsampling filter, (middle trace) impulse response of Adobe ‘Audition1.5’ downsampling filter followed by the Arcam reconstruction filter,and (bottom trace) impulse response of Adobe ‘Audition 1.5’ downsamplingfilter followed by the 12th-order all-pass filter of FIG. 7A followed bythe Arcam reconstruction filter;

FIG. 8 shows group delays of (solid line) the 12th order all-pass filterof FIG. 7A and (dashed line) the 3rd order all-pass filter of FIG. 6A.The vertical axis is calibrated in sample periods at the 88.2 kHzsampling rate;

FIG. 9 shows group delays in sample periods of (full line) a single zeroat z=−1.5, i.e. at radius 1.5 and frequency 22.05 kHz, and (dashedlines) conjugate pairs of zeros at radius 1.5 and frequencies of 20 kHz,16 kHz and 11.025 kHz;

FIG. 10A shows the poles and most of the zeroes of the all-pass filterof FIG. 7A modified for operation at a higher sample rate. Two of thezeroes are outside the range of the plot; and,

FIG. 10B shows the group delay (τ,) in sample periods of the all-passfilter of FIG. 10A when operated at a sample rate of 88.2 kHz.

DETAILED DESCRIPTION

FIG. 1 shows an example recording and reproduction chain in which asound is captured by a microphone 1, converted to digital form by ananalogue-to-digital-converter (ADC) 2 and the resulting signal stored inan archive 3. At some later time the signal is retrieved from thearchive and may pass through a sample rate converter (SRC) 4 and furtherprocessing (P1) 5 before being distributed 6 to listeners either via aphysical medium such as Compact Disc (CD) or by an intangible mediumsuch as radio broadcasting or Internet transmission.

The listener's equipment 7, 8, 9 includes a digital to analogueconverter (DAC) 8 and a transducer 9 such as a headphone or loudspeaker,and optionally further processing (P2) 7.

As will be described later, processing according to the invention may beprovided either as P1 in the mastering equipment 5 or as P2 in thelistener's receiving equipment 7. In both cases, pre-rings generated bythe ADC2 or the SRC 4 or by the listener's DAC 8 will be treated. Insome implementations, processing according to the invention may beprovided at both locations. Furthermore, in some embodiments, processingaccording to the invention may be provided before the SRC, if present,or even before the Archive.

The CD uses a sample rate of 44.1 kHz and throughout the 1980s and 1990smany companies operated the whole recording chain at 44.1 kHz, alsoarchiving at 44.1 kHz so that the SRC 4 was not used. More recentlythere has been a tendency to run the ADC and the archive at a higherrate such as 44.1 kHz, 88.2 kHz, 176.4 kHz, 192 kHz, or even 2.8224 MHzfor 1-bit ‘DSD’ recording, thus necessitating the sample rate converter4, which can be either a separate piece of hardware or part of asoftware Digital Audio Workstation (DAW).

Sample rate conversion has a strong potential to generate pre-responsesbecause of the necessary filtering. This problem is not evaded byrunning the whole chain at 44.1 kHz, for most commercial ADCs thatfurnish a 44.1 kHz output will operate internally at a higher frequencyand then use a sample rate conversion process to provide the desiredoutput sample rate.

Diverse architectures are known for sample rate conversion, the choicedepending on factors such as whether the frequencies involved are in asimple integer ratio such as 2:1 or a more ‘difficult’ ratio such as48:44.1. Alias-free downsampling to 44.1 kHz however always requires alow-pass filter that cuts quite sharply above 20 kHz. The requirementson the shape of the filter are not critically dependent on the samplingfrequency of the source signal. This is true also for upsampling to anarbitrary new sample rate. Thus both downsampling andupsampling/reconstruction generate a requirement for a digital low-passfilter known as an ‘antialias’ filter when downsampling or as a‘reconstruction’ filter when upsampling. The technical requirements forthe two filters are not necessarily very different.

Opinion is divided on whether, when downsampling audio to 44.1 kHz orupsampling from 44.1 kHz, the low-pass filter should provide asubstantial ‘stop-band’ attenuation such as 90 dB at 22.05 kHz orwhether it acceptable to use a filter such as a ‘half-band’ operating at88.2 kHz and configured to provide 6 dB attenuation at 22.05 kHz andfull attenuation by 24.1 kHz. Historically, it was usual to make thefilter's transition band as wide as was considered acceptable in orderto minimise the number of taps in a hardware transversal (‘FIR’)implementation. The transition band was thus about 2 kHz wide, forexample from 20 kHz to 22.05 kHz, or alternatively about 4 kHz wide, forexample from 20 kHz to 24.1 kHz. More recent software implementationshave provided much narrower transition bands, for example the recent‘Adobe Audition CS 5.5’ DAW offers SRC facilities having a transitionband about 100 Hz wide, starting about 75 Hz below the Nyquistfrequency.

Perhaps more typical is the earlier ‘Adobe Audition 1.5’ DAW whichoffers a filter having a transition band about 500 Hz wide, starting at21.5 kHz. Many commercially issued recordings exhibit a near-Nyquistnoise spectrum that suggests that a filter such as this may have beenused at some stage in the processing. FIG. 2 shows the Adobe filter'stransition band and also the transition band of the analogue output of awell-regarded ‘universal’ disc player from Arcam when playing a 44.1 kHzCD. The Arcam response is about 6 dB down at the Nyquist frequency,suggesting that an initial 1:2 upsampling may have been performed usinga ‘half-band’ filter. Other plots of transition bands are shown in the“Sample Rate Conversion Comparison Project”, currently accessible athttp://infinitewave.cakesources.htm.

The impulse responses of the Adobe and Arcam filters are shown in FIG.3A, each having a pre-ring and a post-ring approximately at the Nyquistfrequency. The Adobe filter has the longer pre- and post-responses asmight be expected from its narrower transition band. Close examinationreveals that the Arcam response is essentially zero outside the regionbounded by the two vertical lines, suggesting that a first upsamplingfrom 44.1 kHz to 88.2 kHz has been performed using an FIR filter with aspan of about 107 sample periods at 88.2 kHz.

The Adobe plot is in fact the output of ‘Adobe Audition 1.5’ whenupsampling a single impulse in a 44.1 kHz stream to 88.2 kHz, with the“Pre/Post Filter” and “Quality=999” options selected. Investigationreveals that the same filter is used internally when Audition is used todownsample from 88.2 kHz to 44.1 kHz. In the far ‘tail’ of thepre-response, FIG. 3B, we see a ‘beating’ effect and on taking awindowed Fourier transform of the tail, FIG. 3C we see two distinctfrequencies, 21.5 kHz and 22.05 kHz, corresponding approximately to thetwo edges of the transition band.

To remove the Audition filter's pre-ring a double-notch filter mighttherefore be indicated but this would be specific to the Audition 1.5SRC. We desire a more general method since a music archive may contain44.1 kHz recordings made and/or downsampled using diverse and possiblyunknown equipment.

Pre-response Suppression by Filtering

Assuming pre-responses may have energy in the range 20 kHz-22.05 kHz,one approach is to attenuate this frequency range. A third order IIRfilter having the following z-transform response:

$\frac{\left( {{1.0443\mspace{11mu} z} + 0.9576} \right)\left( {{1.0447\mspace{11mu} z^{2}} + {1.9309\mspace{11mu} z} + 0.9572} \right)}{\left( {{1.2039\mspace{11mu} z} + 0.8307} \right)\left( {{1.2096\mspace{11mu} z^{2}} + {1.8335\mspace{11mu} z} + 0.8267} \right)}$

attenuates the region 20 kHz-22.05 kHz by 20 dB when operated at a 44.1kHz sample rate. This IIR filter has poles (crosses) and zeroes(circles) as shown in FIG. 4A and the frequency response shown in FIG.4B. FIG. 4C shows the Arcam response to a single impulse and to animpulse pre-processed using the above filter. It will be seen that theprocessing has reduced the pre-response significantly, at the expense ofa larger post-response and a frequency response droop of 1 dB at 18 kHz.

According to the invention, the pre-responses may be further reduced byreplacing the minimum-phase filter shown above by the correspondingmaximum-phase filter, as follows:

$\frac{\left( {1.0443 + {0.9576\mspace{11mu} z}} \right)\left( {1.0447\; + 1.9309\; + {0.9572\mspace{11mu} z^{2}}} \right)}{\left( {{1.2039\mspace{11mu} z} + 0.8307} \right)\left( {{1.2096\mspace{11mu} z^{2}} + {1.8335\mspace{11mu} z} + 0.8267} \right)}$

This filter has the same poles but with zeroes outside the unit circle,as shown in FIG. 5A. The frequency response is unchanged from theresponse shown in FIG. 4B. FIG. 5B compares the two responses on a 10×expanded vertical scale, showing that the maximum-phase filter reducesthe downswing immediately prior to the main impulse by 4 dB relative tothe minimum-phase filter and it reduces the other pre-responses by 6 dBor more.

With zeroes outside the unit circle, it is now possible to adjust thepoles inside the unit circle so as to create an all-pass filter:

$\frac{\left( {1.0443 + {0.9576\mspace{11mu} z}} \right)\left( {1.0447\; + {1.9309\mspace{11mu} z}\; + {0.9572\mspace{11mu} z^{2}}} \right)}{\left( {{1.0443\mspace{11mu} z} + 0.9576} \right)\left( {{1.0447\mspace{11mu} z^{2}} + {1.9309\mspace{11mu} z} + 0.9572} \right)}$

whose poles and zeroes are shown in FIG. 6A. This filter has a flatfrequency response from zero to the Nyquist frequency, a property thatsome authorities would consider highly desirable. FIG. 6B shows thatthis filter is able to reduce pre-responses significantly even though itprovides no attenuation at pre-responses frequencies.

More powerful suppression of pre-responses is provided by a 12th orderall-pass filter, as follows:

$\left( \frac{1 + {1.84\mspace{11mu} z} + {0.85\mspace{11mu} z^{2}}}{z^{2} + {1.84\mspace{11mu} z} + 0.85} \right)\left( \frac{1 + {1.81\mspace{11mu} z} + {0.84\mspace{11mu} z^{2}}}{z^{2} + {1.81\mspace{11mu} z} + 0.84} \right)\left( \frac{1 + {1.75\mspace{11mu} z} + {0.81\mspace{11mu} z^{2}}}{z^{2} + {1.75\mspace{11mu} z} + 0.81} \right) \times \left( \frac{1 + {1.33\mspace{11mu} z} + {0.48\mspace{11mu} z^{2}}}{z^{2} + {1.33\mspace{11mu} z} + 0.48} \right)\left( \frac{1 + {0.80\mspace{11mu} z} + {0.19\mspace{11mu} z^{2}}}{z^{2} + {0.80\mspace{11mu} z} + 0.19} \right)\left( \frac{1 + {0.24\mspace{11mu} z} + {0.02\mspace{11mu} z^{2}}}{z^{2} + {0.24\mspace{11mu} z} + 0.02} \right)$

whose poles and zeroes are shown in FIG. 7A.

Referring to FIG. 7B, the top trace shows the impulse response of the‘Adobe Audition 1.5’ filter alone while the middle trace folds in theresponse of the Arcam FMJ DV139 player in an attempt to model a signalchain such as that shown in FIG. 1. The rings far from the central peakare attributable to the sharp transition band of the Audition filter andfolding in the Arcam response reduces them slightly because of its ˜6 dBattenuation at the Nyquist frequency, although the reduction is toosmall to be visible in FIG. 7B.

The bottom trace of FIG. 7B includes the effect of processing the 44.1kHz signal with the 12th order all-pass filter above. The pre-responseshave been almost completely removed. This processing has been found toprovide a substantial audible improvement on many commercially-issuedrecordings.

FIG. 8 shows the group delays of the all-pass filters of FIG. 6A (3rdorder) and FIG. 7A (12th order). Recalling from FIG. 3C that thespectral energy of the pre-responses lies mainly above 20 kHz, the plotsof FIG. 8 strongly suggest that the action of these all-pass filters isto delay the pre-responses, thus converting them into post-responses.

To measure pre-response delay a reference is needed, since a modestdelay of the total signal does not affect the audio quality. One mayconjecture that the ear may use as reference the highest peak in afiltered impulse response or a filtered envelope response. In practiceit is found that non-mimimum-phase zeroes each having a larger groupdelay in the vicinity of 20 kHz than at low audio frequencies arehelpful. We note that group delay at a frequency of 0 Hz is well-definedmathematically: the group delay versus frequency of non-mimimum-phasezeroes having various frequencies over the range 11.025 kHz-22.1 kHz areplotted in FIG. 9. It will be seen that the group delay near 0 Hz isnegative.

Referring again to FIG. 7A, it can be conjectured that the twopole-pairs closest to the origin, namely at −0.12±0.06 i and −0.4±0.16 i(where 1=√1), together with their corresponding zero-pairs at thereciprocal positions:

1/(−0.12±0.06 i)=−6.46±3.43 i

and

1/(−0.4±0.16 i)=−2.15±0.87 i

are contributing little to the group delay near 20 kHz relative to groupdelay at low audio frequencies. Calculation confirms that indeed thesefour zeroes and four poles can be deleted while affecting the saidrelative group delay by only 5% but saving 33% in filter complexity.

Thus in the case of all-pass filters, it is the poles whose real part ismore negative than −0.5 together with their corresponding zeroes thatare most helpful in delaying pre-responses close to the Nyquistfrequency. In the case of filters that are not all-pass, it is thezeroes that are important since a zero can provide helpful attenuationeven if there is no corresponding pole. Thus in general, it is thezeroes whose reciprocals lie inside the unit circle and whose real partsare more negative than −0.5 that are most helpful in reducingpre-responses.

In some cases it is possible to deduce the presence of anon-minimum-phase zero in a filtering apparatus by feeding in asine-wave with an exponentially rising envelope. For example, in thecase of the filter represented in FIG. 6A, a sine-wave at a frequency of20.2 kHz with envelope increasing by a factor of 1.045 on each sampleperiod would theoretically produce zero output because of the zero at−1.0086+0.2723 i.

Of course, such a test signal must have a restricted duration in ordernot to provoke overload and care must be taken that processing delay isnot mistaken for attenuation. A suitable test signal might start at avery low amplitude and contain an impulse as a time reference at the endof the increasing sine-wave. The test could include a comparison of theresponse to that signal with the response to a sine-wave at the samefrequency but with constant amplitude. However, it is not practical totest for zeroes that are far outside the unit circle in this way andthere may also be signal-to-noise difficulties in the case of zeroesthat are extremely close to other zeroes. In difficult cases one mayalternatively capture the impulse response of the apparatus to highprecision using a technique such as chirp excitation, and then apply aroot-finding algorithm to the impulse response.

In the situation depicted in FIG. 1, processing according to theinvention may be placed either as P1 in the mastering equipment 5 or asP2 in the listener's receiving equipment 7. In both cases, pre-ringsgenerated by the SRC 4 or by the listener's DAC 8 will be treated. FIG.7B provides a demonstration that pre-rings from both devices can beeffectively suppressed in a single operation. When new recordings arereleased it would be obvious to place the treatment at P1 for thebenefit of all listeners. Placement at P2 is however of value tolisteners who may already have a collection of media 6 containingrecordings that have not been treated.

The treatment has also been found useful for ‘hi-res’ recordings at asample frequency such as 96 kHz which may contain pre-rings havingfrequencies closer to 48 kHz. The same filter architecture andcoefficients have been used, but clocked at 96 kHz so that the largegroup delay is achieved at frequencies in the range 44 kHz to 48 kHz.

Separately from the above, it is sometimes required to treat a signalthat has already been upsampled: for example there is evidence that somenominally 88.2 kHz or 96 kHz commercially available recordings have beenupsampled from 44.1 kHz or 48 kHz respectively, thereby containingpre-responses just above 20 kHz. In these cases we must distinguishbetween the sampling frequency of the signal presented for treatment anda ‘reference’ sampling frequency which relates to the process thatcreated, or will subsequently create, the pre-rings it is desired totreat. Similar care is needed over the ‘z-transform’: for implementationpurposes ‘z’ must represent a time advance of one sample of the signalpresented for processing, but the criterion previously discussedrelating to the positions of zeroes assumes a ‘z’ that represents onesample period of the process that produced or will produce apre-response.

For the case where the reference sampling frequency is one-half of thesignal's sampling frequency, an appropriate modification to theimprovement filters already presented is to replace z by z² throughout,and hence z² is replaced by z⁴. The poles and zeroes shown in FIG. 7Aare thereby replaced by those shown in FIG. 10A, and if the signal'ssampling rate is 88.2 kHz then the group delay shown as the solid tracein FIG. 8 is extended by reflection about 22.05 kHz as shown in FIG.10B.

The filters thus modified could alternatively be implemented by separateprocessing of substreams consisting of odd samples and even samplesrespectively, and this may be more economical.

These possibilities are not exhaustive, and although the processing willbe performed digitally, it is not excluded that analogue media mayintervene. For example, the archive 3 in FIG. 1 may be a library ofanalogue tapes, some of which may contain pre-responses because digitaleffects units operating internally at 44.1 kHz have been used to processthe signal. As long as analogue media can be assumed linear, processingat the mastering stage 5 will be just as effective in suppressing thesepre-responses as in an all-digital system.

1. A method for reducing the audible effect of a pre-response having energy at a pre-response frequency, the method comprising: introducing group delay at the pre-response frequency by filtering a digital audio signal using a digital non-minimum-phase filter having a z-transform response that includes a zero lying outside the unit circle whose phase response is not linearised by a zero at a reciprocal position inside the unit circle.
 2. A method for reducing the audible effect of a pre-response having energy at a pre-response frequency, the method comprising: introducing group delay at the pre-response frequency by filtering a digital audio signal using a digital non-minimum-phase filter having a z-transform response that includes a zero lying outside the unit circle, wherein the zero is selected to create a greater group delay at the pre-response frequency than at a frequency of 0 Hz.
 3. A method according to claim 1, wherein the digital audio signal contains the pre-response prior to the filtering.
 4. A method according to claim 1, wherein the filtering preconditions the digital audio signal to reduce a pre-response that will be generated in a subsequent upsampling process.
 5. A method according to claim 1, wherein the pre-response frequency lies within 20% of a reference Nyquist frequency equal to one half of a reference sampling frequency that is less than or equal to the sampling frequency of the digital audio signal.
 6. A method according to claim 5, wherein the z-transform response of the filter has at least three zeroes lying outside the unit circle, each selected such that it has a z-plane reciprocal whose real part is more negative than minus 0.5, wherein z represents a time advance of one sample at the reference sampling frequency.
 7. A method according to claim 5, wherein the reference sampling frequency is a sampling frequency of a process that produced the pre-response.
 8. A method according to claim 5, wherein the reference sampling frequency is 44.1 kHz or 48 kHz.
 9. A method according to claim 5, wherein the reference sampling frequency is the sampling frequency of the digital audio signal.
 10. A method according to claim 5, wherein the reference sampling frequency is one half of the sampling frequency of the digital audio signal.
 11. A method according to claim 5, wherein the pre-response frequency is not greater than 60% of a signal Nyquist frequency equal to one half of the sampling frequency of the digital audio signal, and wherein the z-transform response of the filter includes a further zero lying outside the unit circle and contributing a group delay greater at the signal Nyquist frequency than at the pre-response frequency.
 12. A method according to claim 1, wherein the z-transform response of the filter also includes a pole lying inside the unit circle at a reciprocal position to the zero, the pole and zero together selected to create an all-pass factor in the transfer function of the filter.
 13. A method according to claim 1, wherein the z-transform response of the filter comprises one or more zeroes and one or more poles configured such that the combination of zeroes and poles provides an amplitude response flat within 1 dB over the frequency range 0 to 16 kHz.
 14. A method according to claim 1, wherein the zero is selected to create a greater group delay at the pre-response frequency than at a comparison frequency lower than the pre-response frequency.
 15. A method according to claim 14, wherein the group delay at the pre-response frequency exceeds the group delay at the comparison frequency by at least ten cycles at the pre-response frequency.
 16. A method according to claim 14, wherein the comparison frequency is less than or equal to 500 Hz.
 17. A method according to claim 16, wherein the comparison frequency is 0 Hz.
 18. A method according to claim 1, wherein the group delay introduced at the pre-response frequency exceeds by at least ten cycles at the pre-response frequency the time interval from the start of an impulse response of the filter to a sample thereof having the largest absolute magnitude.
 19. A mastering processor adapted to receive a first digital audio signal and to furnish a second digital audio signal for distribution, wherein the mastering processor is configured to reduce the audible effect of a pre-response on a signal rendered from the second signal for auditioning by a listener by introducing group delay at the pre-response frequency by filtering a digital audio signal using a digital non-minimum phase filter having a z-transform response that includes a zero lying outside the unit circle whose phase response is not linearized by a zero at a reciprocal position inside the unit circle, or whose zero is selected to create a greater group delay at the pre-response frequency than at a frequency of 0 Hz.
 20. (canceled)
 21. (canceled)
 22. A non-transitory computer readable medium having stored therein instructions that when executed cause a computer to perform a method for reducing the audible effect of a pre-response having energy at a pre-response frequency, comprising: Introducing a group delay at the pre-response frequency by filtering a digital audio signal using a digital non-minimum phase filter having a z-transform response that includes a zero lying outside the unit-circle whose phase response is not linearized by a zero at a reciprocal position inside the unit circle. 